TA12 最近の更新履歴 Econometrics Ⅰ 2016 TA session

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TA session# 12

Jun Sakamoto

July 20,2016

Endogenous Problem 1

2

Method of moment 3

3

IV; Instrumental variable methods 3

4

2SLS; 2 steps least square 4

5

Examples of IV and 2SLS 7

1 Endogenous Problem

If relation of variables and error are not orthgonal, then a crirical problems are happen in analysis. 1. Model

We consider models of demand function and supply function as below.

P^D= α^D+ β^DQ+ ǫ^D P^S = α^S+ β^SQ+ ǫ^S

E[ǫ^D] = E[ǫ^S] = 0, V ar[ǫ^D] = σ_d², V ar[ǫ^S] = σ²_s, Cov[ǫ^D, ǫ^S] = 0 We can observe prices only at equiribrium. So sample is P^D= P^S = P . Thus Q is

Q=^(α

D−α^S^)+(ǫ^D−ǫ^S⁾ β^S_−β^D

We consider the covariance of Q and error.

Cov[Q, ǫ^S] = E[^(α

D−α^S^)+(ǫ^D−ǫ^S⁾ β^S_−β^D ^ǫ

S_{] =} ^σ²s

β^S_−β^D

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We can understand that Q and error are not orthgonal. Then, what happen OLS estimator? V ar[Q] and Cov[P, Q] are

V ar[Q] = E[(^(α

D−α^S^)+(ǫ^D−ǫ^S⁾ β^S_−β^D ⁾

2_]

=_(β^σ^S²^d_−β^+σ^D²^s₎2

Cov[P, Q] = ^β

SσD²+β^DσS²

(β^S_−β^D)²

β^sis ^Cov[P,Q]_{V ar[Q]} . So we can get

β^s= ^β

Sσ²d+β^Dσ²s

σ_d²+σ²s

= θβ^S_{+ (1 − θ)β}^D θ=_σ2^σ^d²

d+σ²s

So OLS estimator of β is weighted average of β^S and β^D. Thus β has neither unbiasedness nor consistency without θ = 0 or θ = 1.

Next we consider the problem of estimated error.

Y = Xβ + u Z = X + e

We can not observe X and Z is obtained. (For simplicity, X and Z are vector, and β is scalar)Then we estimate a model as below.

Y = Zβ + r Then, this model can be transform as below.

Y = (Z − e)β + r Y = Zβ + (u − eβ) So error is r = (u − eβ). Thus E[Zu] is

E[Xu] = E[(X + e)(u − eβ)] = −βV [e] 6= 0

Thus Z is endogenous variable. Then, OLSE is not unbiaseed and consistent.Because normal OLSE is

E[ ˆβ] = β + E[(X^′X)⁻¹X^′u] E[(X^′X)⁻¹X^′u] is not usually 0. Next we check consistency.

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βˆ_{− β = (}¹_{n i}xix^′_i)^{−1 1}_{n i}xiui

= Q⁻¹_xxQxu

Then by the LLN

1

n i^xⁱûⁱ−→^pÊ[xⁱûⁱ] 6= 0 Thus

plimQ⁻¹_xxQxu_{6= 0}

Hence, β is not consistent.

2 Method of moment

Moment condition is satisfied as below

E[g(X; θ)] = 0

where g() is function vector with dimensional k. X = X1, X2, ..., Xn is sample. Then moment estimator θ is defined as the paramater which satisfies below equation.

1

n^Σⁱg(X; θ) = 0.

3 IV; Instrumental variable methods

Some regression model has endgenous problem. Let z correlate with explanatory variable x and orthogonal with error u. Then we can estimate consistent paramator by IV.(z is called as instrumental variable.) We consider a regression model as below.

y= Xβ + u E[zixi] = Mzxis regular.

E[ziui] = 0

ui, xi, zi has forth order moment. IV estimator ˆβIV is obtained by method of moment.

E[ziui] = E[zi(yi_{− x}^′iβ)] = 0 is assumed.(moment condition) Moment estimator is

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1

n^Σⁱ^[zⁱ^(yⁱ− x^′i^{β)] = 0}

βˆIV = [Σix^′_i]⁻¹Σizixi

= (Z^′X)⁻¹Z^′Y Consistency

βˆIV = (Z^′X)⁻¹Z^′Y (Z^′X)⁻¹Z^′(Xβ + u)

= β + (Z^′X)⁻¹Z^′u

= β + (¹_nΣizix_i^′)^{−1 1}_nΣiziui

By assumption and LLN,

1

n^Σⁱ^zⁱ^x^′i→^p^E[zⁱ^xⁱ^{] = M}^zx^<∞

1

n^Σⁱ^zⁱûⁱ→^pÊ[zⁱûⁱ^{] = 0} So we can get,

βˆIV _→pβ

Asymptotic normality

√_{n( ˆ}_β

IV _{− β) =}^√n(Z^′X)⁻¹Z^′u

= (_n¹Σizix_i^′)^{−1 1}_nΣiziui

We use E[ziui] = 0 and V [ziui] = E[u²_iziz_i^′] = S < ∞ from assumption. Then by CLT and normality

√_{n( ˆ}

βIV _{− β) →}d M_xx⁻¹N(0, S)

= N (0, M_xx⁻¹SM_xx⁻¹)

4 2SLS; 2 steps least square

We consider the case of multiple endogenous variables, exogenous variable and IV.

y= X¹β¹+ X²β²+ u

where X¹:matrix of endogenous variables, X²:exogenous variable and Z:IV. l is a number of IV and k is a number of endogenous variables. We can consider 3 cases.

l > k:over identified l= k:just identified

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Now we consider the case of over identified. Thus if IV are more than endogenous variables, Then we use 2SLS. First we estimate a below equation.

X¹= Z¹δ¹+ X²δ²+ v

We generate predict values ˆX¹ by parameters of above equation. This equation is called as reduced form. We use this values in second step regression as below.

y= ˆX¹β¹+ X²β²+ u

Estimator of above equation is same as IV estimator which employ ˆX¹and X²as IV. Next we define below.

X:A matrix of all variables.((endogenous variables)+(exogenous variables)) Z:A matrix of all exogenous variables.

Xˆ:A matrix of ( ˆX¹, X²) Predict value ˆX is obtained by,

Xˆ = Z(Z^′Z)⁻¹Z^′X Because coefficients of reduced form are written as below.

δ= (Z^′Z)⁻¹Z^′X Thus ˆβ2SLS is

βˆ2SLS = ( ˆX^′X^ˆ)⁻¹X^ˆ^′y

= ((Z(Z^′Z)⁻¹Z^′X)^′Z(Z^′Z)⁻¹Z^′X)⁻¹(Z(Z^′Z)⁻¹Z^′X)⁻¹y

= (X^′Z(Z^′Z)⁻¹Z^′Z(Z^′Z)⁻¹Z^′X)⁻¹X^′Z(Z^′Z)⁻¹Z^′y

= (X^′Z(Z^′Z)⁻¹Z^′X)⁻¹X^′Z(Z^′Z)⁻¹Z^′y Next we define some assumption to confirm asymptotic properties of 2SLS.

E[zix^′_i] = Mzxis regular. E[z^′z] = Mzz is regular.

E[ziui] = 0

All variables have forth order moment.

Consistency

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βˆ2SLS = (X^′Z(Z^′Z)⁻¹Z^′X)⁻¹X^′Z(Z^′Z)⁻¹Z^′y

= (X^′Z(Z^′Z)⁻¹Z^′X)⁻¹X^′Z(Z^′Z)⁻¹Z^′(Xβ + u)

= β + (X^′Z(Z^′Z)⁻¹Z^′X)⁻¹X^′Z(Z^′Z)⁻¹Z^′u

= β + (_n¹Σixiz^′_i(_n¹Σiziz_i^′)^{−1 1}_nΣizix_i^′)^{−1 1}_nΣixiz_i^′(¹_nΣiziz^′_i)^{−1 1}_nΣiziui

By LLN and assumption,

1

n^Σⁱ^xⁱ^zi^′→^p^E[xⁱ^z^′i^{] = M}zx^′ ^<∞

1

n^Σⁱ^zⁱ^z^′i→^p^E[zⁱ^z^′i^{] = M}^zz^<∞

1

n^Σⁱ^zⁱ^x^′i→^p^E[zⁱ^x^′i^{] = M}^zx^<∞

1

n^Σⁱ^zⁱûⁱ→^pÊ[zⁱûⁱ^{] = 0} By the continuous mapping theorem,

(_n¹Σixiz^′_i(_n¹Σiziz_i^′)^{−1 1}_nΣizix^′_i)⁻¹_→p(Mzx^′ ^Mzz⁻¹^Mzx_{) < ∞}

(_n¹Σiziz_i^′)⁻¹_→pE[ziz_i^′] = Mzz⁻¹^<∞ So, we can get,

βˆ2SLS_→pβ

Asymptotic normarity

√_{n( ˆ}

β2SLS_{− β) = (}¹_nΣixiz_i^′(¹_nΣiziz_i^′)^{−1 1}_nΣizix_i^′)^{−1 1}_nΣixiz_i^′(_n¹Σiziz^′_i)^{−1 1}√_nΣiziui

By the LLN,

(¹_nΣixiz_i^′(¹_nΣiziz^′_i)^{−1 1}_nΣizix^′_i)⁻¹ _→p(M_zx^′ M_zz⁻¹Mzx)⁻¹

1

n^Σⁱ^zⁱ^x^′i →^p ^M^zx 1

n^Σⁱ^zⁱ^z

i′ →^p ^E[zⁱ^zi^′] = Mzz⁻¹

E[ziui] = 0, E[u²_iziz_i^′] < inf ty and by the CLT,

√1_nΣiziui_→dN(0, E[u²_iziz_i^′])

= N (0, Ω) Thus, by the CLT,

√_{n( ˆ}_β

2SLS− β) →^d^(Mzx^′ ^Mzz⁻¹^Mzx)⁻¹MzxM_zz⁻¹N(0, Ω)

= N (0, (M^′ M⁻¹M )⁻¹M M⁻¹ΩM⁻¹M (M^′ M⁻¹M )⁻¹)

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5 Examples of IV and 2SLS

We consider a under example.

*IV example*

y= βX + u X = δz + γu + e

Parameters are set as β = 1.5, δ = 2, γ = 0.8. Also z ∼ U(−5, 5), u ∼ N(0, 1), e ∼ N(0, 1) Experiment steps

1.We generate samples z, u and e. 2.y and X made from above model. 3.We estimate ˆβIV and ˆβOLS.

4. ˆβIV and ˆβOLS are obtained by repeating above steps 10000 times.

*2SLS example*

G.D.Pietro(2012)”Does studying abroad cause international labor mobility? Evidence from Italy” Economic Letters 117 pp632-pp635

This paper confirm relationship of studying abroad and international labor mobility.Several studies(Parey and Waldinger etc)shows that there is a positive relationship between international student mobility and the desicion to work abroad following graduation.

However that studies faced omitted variable bias. Thus, this paper use 2SLS method.

Model

The following baseline specification can be used to investigate the effect of studying abroad on the decision to work in a foreign country.

Yijk= β0+ β1studyabroadijk+ β2Xijk+ β3Dj+ β4Uk+ ǫijk

Y:takes the 1 if he/she works abroad and 0 other wise. studyabroad:takes the 1 if he/she study abroad and 0 otherwise.

X, D, U:control variables. And reduced form is

studyabroadijk= α0+ α1ex− studyabroadîjk^{+ α}²^Xîjk^{+ α}³^D^j^{+ α}⁴Û^k^{+ ǫ}îjk ex− studyabroad is a measure of students’ exprosure to study abroad programs.

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Figure 1:

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